Computer simulation of fluid flow and acoustic behavior

ABSTRACT

A computer-implemented method for simulating flow and acoustic interaction of a fluid with a porous medium includes simulating activity of a fluid in a first volume adjoining a second volume, the activity of the fluid in the first volume being simulated so as to model movement of elements within the first volume and using a first model having a first set of parameters, simulating activity of the fluid in the second volume occupied by the porous medium, the activity in the second volume being simulated so as to model movement of elements within the second volume and using a second model having a second set of parameters, and simulating movement of elements between the first volume and the second volume at an interface between the first volume and the second volume.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of and claims benefit to U.S.application Ser. No. 13/292,844, filed on Nov. 9, 2011. The applicationis incorporated by reference in its entirety.

TECHNICAL FIELD

This description relates to computer simulation of physical processes,such as fluid flow and acoustics.

BACKGROUND

High Reynolds number flow has been simulated by generating discretizedsolutions of the Navier-Stokes differential equations by performinghigh-precision floating point arithmetic operations at each of manydiscrete spatial locations on variables representing the macroscopicphysical quantities (e.g., density, temperature, flow velocity). Anotherapproach replaces the differential equations with what is generallyknown as lattice gas (or cellular) automata, in which themacroscopic-level simulation provided by solving the Navier-Stokesequations is replaced by a microscopic-level model that performsoperations on particles moving between sites on a lattice.

SUMMARY

In general, this document describes techniques for simulatinginteractions between a fluid and a porous medium at an interface betweenthe fluid and the porous medium. These interactions may be simulated forthe purpose of characterizing acoustic properties of a system includingthe fluid and the porous medium. The volume properties that permitaccurate simulation of flow resistance also may be used to permitaccurate simulation of acoustic resistance. As such, representing aporous medium with flow resistance properties and accurately modelingthe interface between a fluid space and the porous medium may lead toaccurate representations of the acoustic behavior of the porous medium(i.e., the porous medium's ability to attenuate sound). Accuraterepresentation of the acoustic properties may permit improved simulationof physical phenomena such as the acoustic impedance or acousticabsorption of materials in complex systems such as vehicles andbuildings.

In one general aspect, a computer-implemented method for simulating flowand acoustic interaction of a fluid with a porous medium includessimulating activity of a fluid in a first volume adjoining a secondvolume occupied by a porous medium. The activity of the fluid in thefirst volume is simulated so as to model movement of elements within thefirst volume and using a first model having a first set of parameters.In addition, activity of the fluid in the second volume occupied by theporous medium is simulated so as to model movement of elements withinthe second volume and using a second model having a second set ofparameters and differing from the first model in a way that accounts forflow and acoustic properties of the porous medium. Finally, movement ofelements between the first volume and the second volume at an interfacebetween the first volume and the second volume is simulated.

Implementations may include one or more of the following features. Forexample, a computer-accessible memory may be used to store a first setof state vectors for voxels of the first volume and a second set ofstate vectors for voxels of the second volume, with each of the statevectors including entries that correspond to particular momentum statesof possible momentum states at a corresponding voxel. Simulatingactivity of the fluid in the first volume may include performing firstinteraction operations on the first set of state vectors, the firstinteraction operations modeling flow interactions between elements ofdifferent momentum states according to the first model, and performingfirst move operations on the first set of state vectors to reflectmovement of elements to new voxels in the first volume according to thefirst model. Simulating activity of the fluid in the second volume mayinclude performing second interaction operations on the second set ofstate vectors, the second interaction operations modeling flowinteractions between elements of different momentum states according tothe second model, and performing second move operations on the secondset of state vectors to reflect movement of elements to new voxels inthe second volume according to the second model.

Simulating movement of elements between the first volume and the secondvolume at the interface between the first volume and the second volumemay include simulating movement of elements between a region of thefirst volume at the interface and a region of the second volume at theinterface. For example, simulating movement of elements between thefirst volume and the second volume at the interface between the firstvolume and the second volume may include simulating movement of elementsbetween state vectors of the first set of state vectors for voxels inthe first volume at the interface and state vectors of the second set ofstate vectors for voxels in the second volume at the interface.

Movement of elements from the first volume to the second volume may begoverned by a first set of constraints and movement of elements from thesecond volume to the first volume may be governed by a second set ofconstraints that differs from the first set of constraints. The firstset of constraints may permit a fraction of elements oriented to movefrom the first volume to the second volume to actually move from thefirst volume to the second volume, the fraction corresponding to aporosity of the porous medium being simulated, while the second set ofconstraints may permit all elements oriented to move from the secondvolume to the first volume to actually move from the second volume tothe first volume. The second set of parameters of the second model maydiffer from the first set of parameters of the first model in a way thataccounts for a porosity of the porous medium being simulated, or one ormore of acoustic resistance, acoustic absorption, and acoustic impedanceof the porous medium being simulated. The second set of parameters ofthe second model may also differ from the first set of parameters of thefirst model in a way that accounts for one or more of tortuosity,characteristic viscous length, thermal characteristic length, andthermal permeability of the porous medium being simulated.

The elements may include particle distributions or fluxes ofhydrodynamic and thermodynamic properties such as mass fluxes, momentumfluxes, and energy fluxes. Additionally, the elements may includeproperties such as mass, density, momentum, pressure, velocity,temperature, and energy. Moreover, the elements may be associated withany fluid, flow, or thermodynamic related quantity although notexhaustively noted above or explicitly discussed below.

The first volume may include, for example, one or more of an interiorcabin of a vehicle and an exterior of an aircraft, and the second volumemay include static components and surfaces within the interior cabin ofthe vehicle. In a more particular example, the first volume may includeregions surrounding an aircraft landing gear assembly of the exterior ofthe aircraft, and the second volume may include regions betweenindividual components of the aircraft landing gear assembly.

In another general aspect, a computer-implemented method for simulatingflow and acoustic interaction of a fluid with a porous medium includessimulating activity of a fluid in a first volume adjoining a secondvolume occupied by a porous medium, with the activity of the fluid inthe first volume being simulated so as to model movement of elementswithin the first volume and using a first model having a first set ofparameters; simulating activity of the fluid in the second volumeoccupied by the porous medium, with the activity in the second volumebeing simulated so as to model movement of elements within the secondvolume and using a second model having a second set of parameters anddiffering from the first model in a way that accounts for properties ofthe porous medium; and simulating movement of elements between the firstvolume and the second volume at an interface between the first volumeand the second volume by simulating movement of elements between aregion of the first volume at the interface and a region of the secondvolume at the interface. Movement of elements from the first volume tothe second volume is governed by a first set of constraints and movementof elements from the second volume to the first volume is governed by asecond set of constraints that differs from the first set ofconstraints. The first set of constraints permits a fraction of elementsoriented to move from the first volume to the second volume to actuallymove from the first volume to the second volume, the fractioncorresponding to a porosity of the porous medium being simulated.

Implementations may include one or more of the features noted above ordiscussed below.

In other general aspects, the methods and techniques noted above anddescribed below are included in a system for simulating flow andacoustic interaction of a fluid with a porous medium and acomputer-readable data storage medium storing computer-executableinstructions that, when executed, simulate flow and acoustic interactionof a fluid with a porous medium.

The systems and techniques may be implemented using a lattice gassimulation that employs a Lattice Boltzmann formulation. The traditionallattice gas simulation assumes a limited number of particles at eachlattice site, with the particles being represented by a short vector ofbits. Each bit represents a particle moving in a particular direction.For example, one bit in the vector might represent the presence (whenset to 1) or absence (when set to 0) of a particle moving along aparticular direction. Such a vector might have six bits, with, forexample, the values 110000 indicating two particles moving in oppositedirections along the X axis, and no particles moving along the Y and Zaxes. A set of collision rules governs the behavior of collisionsbetween particles at each site (e.g., a 110000 vector might become a001100 vector, indicating that a collision between the two particlesmoving along the X axis produced two particles moving away along the Yaxis). The rules are implemented by supplying the state vector to alookup table, which performs a permutation on the bits (e.g.,transforming the 110000 to 001100). Particles are then moved toadjoining sites (e.g., the two particles moving along the Y axis wouldbe moved to neighboring sites to the left and right along the Y axis).

In an enhanced system, the state vector at each lattice site includesmany more bits (e.g., 54 bits for subsonic flow) to provide variation inparticle energy and movement direction, and collision rules involvingsubsets of the full state vector are employed. In a further enhancedsystem, more than a single particle is permitted to exist in eachmomentum state at each lattice site, or voxel (these two terms are usedinterchangeably throughout this document). For example, in an eight-bitimplementation, 0-255 particles could be moving in a particulardirection at a particular voxel. The state vector, instead of being aset of bits, is a set of integers (e.g., a set of eight-bit bytesproviding integers in the range of 0 to 255), each of which representsthe number of particles in a given state.

In a further enhancement, Lattice Boltzmann Methods (LBM) use amesoscopic representation of a fluid to simulate 3D unsteadycompressible turbulent flow processes in complex geometries at a deeperlevel than possible with conventional computational fluid dynamics(“CFD”) approaches. A brief overview of LBM method is provided below.

Boltzmann-Level Mesoscopic Representation

It is well known in statistical physics that fluid systems can berepresented by kinetic equations on the so-called “mesoscopic” level. Onthis level, the detailed motion of individual particles need not bedetermined. Instead, properties of a fluid are represented by theparticle distribution functions defined using a single particle phasespace, ƒ=ƒ(x,v,t), where x is the spatial coordinate while v is theparticle velocity coordinate. The typical hydrodynamic quantities, suchas mass, density, fluid velocity and temperature, are simple moments ofthe particle distribution function. The dynamics of the particledistribution functions obeys a Boltzmann equation:∂_(t) ƒ+v∇ _(x) ƒ+F(x,t)∇_(v) ƒ=C{ƒ},  Eq.(1)where F(x,t) represents an external or self-consistently generatedbody-force at (x,t). The collision term C represents interactions ofparticles of various velocities and locations. It is important to stressthat, without specifying a particular form for the collision term C, theabove Boltzmann equation is applicable to all fluid systems, and notjust to the well known situation of rarefied gases (as originallyconstructed by Boltzmann).

Generally speaking, C includes a complicated multi-dimensional integralof two-point correlation functions. For the purpose of forming a closedsystem with distribution functions ƒ alone as well as for efficientcomputational purposes, one of the most convenient and physicallyconsistent forms is the well-known BGK operator. The BGK operator isconstructed according to the physical argument that, no matter what thedetails of the collisions, the distribution function approaches awell-defined local equilibrium given by {ƒ^(eq)(x,v,t)} via collisions:

$\begin{matrix}{{C = {{- \frac{1}{\tau}}\left( {f - f^{eq}} \right)}},} & {{Eq}.\mspace{14mu}(2)}\end{matrix}$where the parameter τ represents a characteristic relaxation time toequilibrium via collisions. Dealing with particles (e.g., atoms ormolecules) the relaxation time is typically taken as a constant. In a“hybrid” (hydro-kinetic) representation, this relaxation time is afunction of hydrodynamic variables like rate of strain, turbulentkinetic energy and others. Thus, a turbulent flow may be represented asa gas of turbulence particles (“eddies”) with the locally determinedcharacteristic properties.

Numerical solution of the Boltzmann-BGK equation has severalcomputational advantages over the solution of the Navier-Stokesequations. First, it may be immediately recognized that there are nocomplicated nonlinear terms or higher order spatial derivatives in theequation, and thus there is little issue concerning advectioninstability. At this level of description, the equation is local sincethere is no need to deal with pressure, which offers considerableadvantages for algorithm parallelization. Another desirable feature ofthe linear advection operator, together with the fact that there is nodiffusive operator with second order spatial derivatives, is its ease inrealizing physical boundary conditions such as no-slip surface orslip-surface in a way that mimics how particles truly interact withsolid surfaces in reality, rather than mathematical conditions for fluidpartial differential equations (“PDEs”). One of the direct benefits isthat there is no problem handling the movement of the interface on asolid surface, which helps to enable lattice-Boltzmann based simulationsoftware to successfully simulate complex turbulent aerodynamics. Inaddition, certain physical properties from the boundary, such as finiteroughness surfaces, can also be incorporated in the force. Furthermore,the BGK collision operator is purely local, while the calculation of theself-consistent body-force can be accomplished via near-neighborinformation only. Consequently, computation of the Boltzmann-BGKequation can be effectively adapted for parallel processing.

Lattice Boltzmann Formulation

Solving the continuum Boltzmann equation represents a significantchallenge in that it entails numerical evaluation of anintegral-differential equation in position and velocity phase space. Agreat simplification took place when it was observed that not only thepositions but the velocity phase space could be discretized, whichresulted in an efficient numerical algorithm for solution of theBoltzmann equation. The hydrodynamic quantities can be written in termsof simple sums that at most depend on nearest neighbor information. Eventhough historically the formulation of the lattice Boltzmann equationwas based on lattice gas models prescribing an evolution of particles ona discrete set of velocities v(ε{c_(i), i=1, . . . , b}), this equationcan be systematically derived from the first principles as adiscretization of the continuum Boltzmann equation. As a result, LBEdoes not suffer from the well-known problems associated with the latticegas approach. Therefore, instead of dealing with the continuumdistribution function in phase space, ƒ(x,v,t), it is only necessary totrack a finite set of discrete distributions, ƒ_(i)(x,t) with thesubscript labeling the discrete velocity indices. The key advantage ofdealing with this kinetic equation instead of a macroscopic descriptionis that the increased phase space of the system is offset by thelocality of the problem.

Due to symmetry considerations, the set of velocity values are selectedin such a way that they form certain lattice structures when spanned inthe configuration space. The dynamics of such discrete systems obeys theLBE having the form ƒ_(i)(x+c_(i), t+1)−ƒ_(i)(x,t)=C_(i)(x,t), where thecollision operator usually takes the BGK form as described above. Byproper choices of the equilibrium distribution forms, it can betheoretically shown that the lattice Boltzmann equation gives rise tocorrect hydrodynamics and thermo-hydrodynamics. That is, thehydrodynamic moments derived from ƒ_(i)(x,t) obey the Navier-Stokesequations in the macroscopic limit. These moments are defined as:

$\begin{matrix}{{{{\rho\left( {x,t} \right)} = {\sum\limits_{i}{f_{i}\left( {x,t} \right)}}};}{{{\rho\;{u\left( {x,t} \right)}} = {\sum\limits_{i}{c_{i}{f_{i}\left( {x,t} \right)}}}};}{{{{DT}\left( {x,t} \right)} = {\sum\limits_{i}{\left( {c_{i} - u} \right)^{2}{f_{i}\left( {x,t} \right)}}}},}} & {{Eq}.\mspace{14mu}(3)}\end{matrix}$where ρ, u, and T are, respectively, the fluid density, velocity andtemperature, and D is the dimension of the discretized velocity space(not at all equal to the physical space dimension).

Other features and advantages will be apparent from the followingdescription, including the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 and 2 illustrate velocity components of two LBM models.

FIG. 3 is a flow chart of a procedure followed by a physical processsimulation system.

FIG. 4 is a perspective view of a microblock.

FIGS. 5A and 5B are illustrations of lattice structures used by thesystem of FIG. 3.

FIGS. 6 and 7 illustrate variable resolution techniques.

FIG. 8 illustrates regions affected by a facet of a surface.

FIG. 9 illustrates movement of particles from a voxel to a surface.

FIG. 10 illustrates movement of particles from a surface to a surface.

FIG. 11 is a flow chart of a procedure for performing surface dynamics.

FIG. 12 illustrates an interface between voxels of different sizes.

FIG. 13 is a flow chart of a procedure for simulating interactions withfacets under variable resolution conditions.

FIG. 14 is a schematic view of an exemplary porous media model.

FIG. 15 is a schematic view of exemplary double-sided surfels.

FIG. 16 is a schematic view of an exemplary system for modeling acousticabsorption.

FIGS. 17A and 17B are schematic views of other exemplary porous mediamodels.

FIG. 18 is a schematic view of another exemplary porous media model.

FIG. 19 illustrates absorption coefficients as a function of frequency.

FIGS. 20-22 illustrate absorption coefficients as functions offrequency.

FIG. 23 illustrates an interior of a vehicle.

FIG. 24 illustrates an aircraft landing system.

DESCRIPTION

A. Volumetric Approach to Modeling Acoustic Absorption

Acoustic absorption, i.e., acoustic resistance, acoustic impedance,etc., by porous materials is an important topic in acousticsengineering. At a microscopic scale, the propagation of sound in porousmedia is difficult to characterize because of the topological complexityof the materials. At a macroscopic scale, porous materials with highporosity can be treated as regions of fluid which have modifiedproperties relative to air. Sound propagation in such media can beexpressed in the form of two intrinsic, frequency-dependent, andvolumetric properties of the material: the characteristic impedance andthe complex acoustic wave number. These properties may be modeled indifferent ways. For example, under certain assumptions, a givenvolumetric model for sound propagation in an absorbing material can beput in the form of a locally-reacting, frequency-dependent, compleximpedance at the interface between two different media. Such impedancemodels may be used in approaches such as the Boundary Element Methods(BEM), the Finite Elements Methods (FEM), and the Statistical EnergyAnalysis (SEA) methods, and may be implemented as boundary conditions inthe frequency domain.

For problems involving flow-induced noise, suitable Computational FluidDynamics (CFD) and/or Computational AeroAcoustics (CAA) numericalmethods are non-linear and often time-explicit. For a time-explicitsolution, time-domain surface impedance boundary conditions may allowmodeling of acoustic absorption due to porous materials. However, evenwhen a time-domain surface impedance formulation can be derived,stability and robustness may be challenging problems to overcome.

Another approach, which is described in more detail below, includesmodeling of absorbing materials as volumetric fluid regions, such thatsound waves travel through the material and dissipate via a momentumsink. This is analogous to the method for macroscopic modeling of flowthrough porous media achieved by relating the momentum sink to the flowresistance of the material following Darcy's law. For acousticabsorption modeling, there is the question of how to determine themomentum sink to achieve a desired absorption behavior. If the acousticabsorption is governed (or at least dominated) by the same physicalmechanisms as the flow resistivity, then the same momentum sink behaviorused to achieve the correct flow resistivity for a particular porousmaterial should also achieve the correct acoustic absorption for thatmaterial. This approach may be applicable for any passive andhomogeneous porous material. Moreover, the approach eliminates numericalstability problems since the impedance is realized in a way thatsatisfies passive, causal, and real conditions.

This volumetric modeling approach may be used in conjunction with atime-explicit CFD/CAA solution method based on the Lattice BoltzmannMethod (LBM), such as the PowerFLOW system available from ExaCorporation of Burlington, Mass. Unlike methods based on discretizingthe macroscopic continuum equations, LBM starts from a “mesoscopic”Boltzmann kinetic equation to predict macroscopic fluid dynamics. Theresulting compressible and unsteady solution method may be used forpredicting a variety of complex flow physics, such as aeroacoustics andpure acoustics problems. A porous media model is used to represent theflow resistivity of various components, such as air filters, radiators,heat exchangers, evaporators, and other components, which areencountered in simulating flow, such as through HVAC systems, vehicleengine compartments, and other applications.

A general discussion of a LBM-based simulation system is provided belowand followed by a discussion of a volumetric modeling approach foracoustic absorption and other phenomena and a porous media interfacemodel that may be used to support such a volumetric modeling approach.

B. Model Simulation Space

In a LBM-based physical process simulation system, fluid flow may berepresented by the distribution function values ƒ_(i), evaluated at aset of discrete velocities c_(i). The dynamics of the distributionfunction is governed by Equation 4 where ƒ_(i)(0) is known as theequilibrium distribution function, defined as:

$\begin{matrix}{f_{\alpha}^{(0)} = {w_{\alpha}{\rho\left\lbrack {1 + u_{\alpha} + \frac{u_{\alpha}^{2} - u^{2}}{2} + \frac{u_{\alpha}\left( {u_{\alpha}^{2} - {3u^{2}}} \right)}{6}} \right\rbrack}}} & {{Eq}.\mspace{14mu}(4)}\end{matrix}$This equation is the well-known lattice Boltzmann equation that describethe time-evolution of the distribution function, ƒ_(i). The left-handside represents the change of the distribution due to the so-called“streaming process.” The streaming process is when a pocket of fluidstarts out at a grid location, and then moves along one of the velocityvectors to the next grid location. At that point, the “collisionfactor,” i.e., the effect of nearby pockets of fluid on the startingpocket of fluid, is calculated. The fluid can only move to another gridlocation, so the proper choice of the velocity vectors is necessary sothat all the components of all velocities are multiples of a commonspeed.

The right-hand side of the first equation is the aforementioned“collision operator” which represents the change of the distributionfunction due to the collisions among the pockets of fluids. Theparticular form of the collision operator used here is due to Bhatnagar,Gross and Krook (BGK). It forces the distribution function to go to theprescribed values given by the second equation, which is the“equilibrium” form.

From this simulation, conventional fluid variables, such as mass ρ andfluid velocity u, are obtained as simple summations in Equation (3).Here, the collective values of c_(i) and w_(i) define a LBM model. The

LBM model can be implemented efficiently on scalable computer platformsand run with great robustness for time unsteady flows and complexboundary conditions.

A standard technique of obtaining the macroscopic equation of motion fora fluid system from the Boltzmann equation is the Chapman-Enskog methodin which successive approximations of the full Boltzmann equation aretaken.

In a fluid system, a small disturbance of the density travels at thespeed of sound. In a gas system, the speed of the sound is generallydetermined by the temperature. The importance of the effect ofcompressibility in a flow is measured by the ratio of the characteristicvelocity and the sound speed, which is known as the Mach number.

Referring to FIG. 1, a first model (2D−1) 100 is a two-dimensional modelthat includes 21 velocities. Of these 21 velocities, one (105)represents particles that are not moving; three sets of four velocitiesrepresent particles that are moving at either a normalized speed (r)(110-113), twice the normalized speed (2r) (120-123), or three times thenormalized speed (3r) (130-133) in either the positive or negativedirection along either the x or y axis of the lattice; and two sets offour velocities represent particles that are moving at the normalizedspeed (r) (140-143) or twice the normalized speed (2r) (150-153)relative to both of the x and y lattice axes.

As also illustrated in FIG. 2, a second model (3D−1) 200 is athree-dimensional model that includes 39 velocities, where each velocityis represented by one of the arrowheads of FIG. 2. Of these 39velocities, one represents particles that are not moving; three sets ofsix velocities represent particles that are moving at either anormalized speed (r), twice the normalized speed (2r), or three timesthe normalized speed (3r) in either the positive or negative directionalong the x, y or z axis of the lattice; eight represent particles thatare moving at the normalized speed (r) relative to all three of the x,y, z lattice axes; and twelve represent particles that are moving attwice the normalized speed (2r) relative to two of the x, y, z latticeaxes.

More complex models, such as a 3D−2 model includes 101 velocities and a2D−2 model includes 37 velocities also may be used. The velocities aremore clearly described by their component along each axis as documentedin Tables 1 and 2 respectively.

For the three-dimensional model 3D−2, of the 101 velocities, onerepresents particles that are not moving (Group 1); three sets of sixvelocities represent particles that are moving at either a normalizedspeed (r), twice the normalized speed (2r), or three times thenormalized speed (3r) in either the positive or negative direction alongthe x, y or z axis of the lattice (Groups 2, 4, and 7); three sets ofeight represent particles that are moving at the normalized speed (r),twice the normalized speed (2r), or three times the normalized speed(3r) relative to all three of the x, y, z lattice axes (Groups 3, 8, and10); twelve represent particles that are moving at twice the normalizedspeed (2r) relative to two of the x, y, z lattice axes (Group 6); twentyfour represent particles that are moving at the normalized speed (r) andtwice the normalized speed (2r) relative to two of the x, y, z latticeaxes, and not moving relative to the remaining axis (Group 5); andtwenty four represent particles that are moving at the normalized speed(r) relative to two of the x, y, z lattice axes and three times thenormalized speed (3r) relative to the remaining axis (Group 9).

For the two-dimensional model 2D−2, of the 37 velocities, one representsparticles that are not moving (Group 1); three sets of four velocitiesrepresent particles that are moving at either a normalized speed (r),twice the normalized speed (2r), or three times the normalized speed(3r) in either the positive or negative direction along either the x ory axis of the lattice (Groups 2, 4, and 7); two sets of four velocitiesrepresent particles that are moving at the normalized speed (r) or twicethe normalized speed (2r) relative to both of the x and y lattice axes;eight velocities represent particles that are moving at the normalizedspeed (r) relative to one of the x and y lattice axes and twice thenormalized speed (2r) relative to the other axis; and eight velocitiesrepresent particles that are moving at the normalized speed (r) relativeto one of the x and y lattice axes and three times the normalized speed(3r) relative to the other axis.

The LBM models described above provide a specific class of efficient androbust discrete velocity kinetic models for numerical simulations offlows in both two- and three-dimensions. A model of this kind includes aparticular set of discrete velocities and weights associated with thosevelocities. The velocities coincide with grid points of Cartesiancoordinates in velocity space which facilitates accurate and efficientimplementation of discrete velocity models, particularly the kind knownas the lattice Boltzmann models. Using such models, flows can besimulated with high fidelity.

Referring to FIG. 3, a physical process simulation system operatesaccording to a procedure 300 to simulate a physical process such asfluid flow. Prior to the simulation, a simulation space is modeled as acollection of voxels (step 302). Typically, the simulation space isgenerated using a computer-aided-design (CAD) program. For example, aCAD program could be used to draw an micro-device positioned in a windtunnel. Thereafter, data produced by the CAD program is processed to adda lattice structure having appropriate resolution and to account forobjects and surfaces within the simulation space.

The resolution of the lattice may be selected based on the Reynoldsnumber of the system being simulated. The Reynolds number is related tothe viscosity (v) of the flow, the characteristic length (L) of anobject in the flow, and the characteristic velocity (u) of the flow:Re=uL/v.  Eq.(5)

The characteristic length of an object represents large scale featuresof the object. For example, if flow around a micro-device were beingsimulated, the height of the micro-device might be considered to be thecharacteristic length. When flow around small regions of an object(e.g., the side mirror of an automobile) is of interest, the resolutionof the simulation may be increased, or areas of increased resolution maybe employed around the regions of interest. The dimensions of the voxelsdecrease as the resolution of the lattice increases.

The state space is represented as ƒ_(i)(x,t), where ƒ_(i) represents thenumber of elements, or particles, per unit volume in state i (i.e., thedensity of particles in state i) at a lattice site denoted by thethree-dimensional vector x at a time t. For a known time increment, thenumber of particles is referred to simply as ƒ_(i)(x). The combinationof all states of a lattice site is denoted as ƒ(x).

The number of states is determined by the number of possible velocityvectors within each energy level. The velocity vectors consist ofinteger linear speeds in a space having three dimensions: x, y, and z.The number of states is increased for multiple-species simulations.

Each state i represents a different velocity vector at a specific energylevel (i.e., energy level zero, one or two). The velocity c_(i) of eachstate is indicated with its “speed” in each of the three dimensions asfollows:c _(i)=(c _(i,x) ,c _(i,y) ,c _(i,z)).  Eq.(6)

The energy level zero state represents stopped particles that are notmoving in any dimension, i.e. c_(stopped)=(0, 0, 0). Energy level onestates represent particles having a ±1 speed in one of the threedimensions and a zero speed in the other two dimensions. Energy leveltwo states represent particles having either a ±1 speed in all threedimensions, or a ±2 speed in one of the three dimensions and a zerospeed in the other two dimensions.

Generating all of the possible permutations of the three energy levelsgives a total of 39 possible states (one energy zero state, 6 energy onestates, 8 energy three states, 6 energy four states, 12 energy eightstates and 6 energy nine states.).

Each voxel (i.e., each lattice site) is represented by a state vectorf(x). The state vector completely defines the status of the voxel andincludes 39 entries. The 39 entries correspond to the one energy zerostate, 6 energy one states, 8 energy three states, 6 energy four states,12 energy eight states and 6 energy nine states. By using this velocityset, the system can produce Maxwell-Boltzmann statistics for an achievedequilibrium state vector.

For processing efficiency, the voxels are grouped in 2×2×2 volumescalled microblocks. The microblocks are organized to permit parallelprocessing of the voxels and to minimize the overhead associated withthe data structure. A short-hand notation for the voxels in themicroblock is defined as N_(i) (n), where n represents the relativeposition of the lattice site within the microblock and nε{0, 1, 2, . . ., 7}. A microblock is illustrated in FIG. 4.

Referring to FIGS. 5A and 5B, a surface S (FIG. 3A) is represented inthe simulation space (FIG. 5B) as a collection of facets F_(α):S={F _(α)}  Eq.(7)where α is an index that enumerates a particular facet. A facet is notrestricted to the voxel boundaries, but is typically sized on the orderof or slightly smaller than the size of the voxels adjacent to the facetso that the facet affects a relatively small number of voxels.Properties are assigned to the facets for the purpose of implementingsurface dynamics. In particular, each facet F_(α) has a unit normal(n_(α)), a surface area (A_(α)), a center location (x_(α)), and a facetdistribution function (ƒ_(i)(α)) that describes the surface dynamicproperties of the facet.

Referring to FIG. 6, different levels of resolution may be used indifferent regions of the simulation space to improve processingefficiency. Typically, the region 650 around an object 655 is of themost interest and is therefore simulated with the highest resolution.Because the effect of viscosity decreases with distance from the object,decreasing levels of resolution (i.e., expanded voxel volumes) areemployed to simulate regions 660, 665 that are spaced at increasingdistances from the object 655. Similarly, as illustrated in FIG. 7, alower level of resolution may be used to simulate a region 770 aroundless significant features of an object 775 while the highest level ofresolution is used to simulate regions 780 around the most significantfeatures (e.g., the leading and trailing surfaces) of the object 775.Outlying regions 785 are simulated using the lowest level of resolutionand the largest voxels.

C. Identify Voxels Affected by Facets

Referring again to FIG. 3, once the simulation space has been modeled(step 302), voxels affected by one or more facets are identified (step304). Voxels may be affected by facets in a number of ways. First, avoxel that is intersected by one or more facets is affected in that thevoxel has a reduced volume relative to non-intersected voxels. Thisoccurs because a facet, and material underlying the surface representedby the facet, occupies a portion of the voxel. A fractional factorP_(f)(x) indicates the portion of the voxel that is unaffected by thefacet (i.e., the portion that can be occupied by a fluid or othermaterials for which flow is being simulated). For non-intersectedvoxels, P_(f)(x) equals one.

Voxels that interact with one or more facets by transferring particlesto the facet or receiving particles from the facet are also identifiedas voxels affected by the facets. All voxels that are intersected by afacet will include at least one state that receives particles from thefacet and at least one state that transfers particles to the facet. Inmost cases, additional voxels also will include such states.

Referring to FIG. 8, for each state i having a non-zero velocity vectorc_(i), a facet F_(α) receives particles from, or transfers particles to,a region defined by a parallelepiped G_(iα), having a height defined bythe magnitude of the vector dot product of the velocity vector c_(i) andthe unit normal n_(α) of the facet (|c_(i)n_(i)|) and a base defined bythe surface area A_(α) of the facet so that the volume V_(iα) of theparallelepiped G_(iα) equals:V _(iα) =|c _(i) n _(α) |A _(α)  Eq.(8)

The facet F_(α) receives particles from the volume V_(iα) when thevelocity vector of the state is directed toward the facet (|c_(i)n_(i)|<0), and transfers particles to the region when the velocityvector of the state is directed away from the facet (|c_(i) n_(i)|>0).As will be discussed below, this expression must be modified whenanother facet occupies a portion of the parallelepiped G_(iα), acondition that could occur in the vicinity of non-convex features suchas interior corners.

The parallelepiped G_(iα) of a facet F_(α) may overlap portions or allof multiple voxels. The number of voxels or portions thereof isdependent on the size of the facet relative to the size of the voxels,the energy of the state, and the orientation of the facet relative tothe lattice structure. The number of affected voxels increases with thesize of the facet. Accordingly, the size of the facet, as noted above,is typically selected to be on the order of or smaller than the size ofthe voxels located near the facet.

The portion of a voxel N(x) overlapped by a parallelepiped G_(iα) isdefined as V_(iα)(x). Using this term, the flux Γ_(iα)(x) of state iparticles that move between a voxel N(x) and a facet F_(α) equals thedensity of state i particles in the voxel (N_(i)(x)) multiplied by thevolume of the region of overlap with the voxel (V_(iα)(x)):Γ_(iα)(x)=N _(i)(x)V _(iα)(x).  Eq.(9)

When the parallelepiped G_(iα) is intersected by one or more facets, thefollowing condition is true:V _(iα) =ΣV _(α)(x)+ΣV _(iα)(β)  Eq.(10)

where the first summation accounts for all voxels overlapped by G_(iα)and the second term accounts for all facets that intersect G_(iα). Whenthe parallelepiped G_(iα) is not intersected by another facet, thisexpression reduces to:V _(iα) =ΣV _(iα)(x).  Eq.(11)

D. Perform Simulation

Once the voxels that are affected by one or more facets are identified(step 304), a timer is initialized to begin the simulation (step 306).During each time increment of the simulation, movement of particles fromvoxel to voxel is simulated by an advection stage (steps 308-316) thataccounts for interactions of the particles with surface facets. Next, acollision stage (step 318) simulates the interaction of particles withineach voxel. Thereafter, the timer is incremented (step 320). If theincremented timer does not indicate that the simulation is complete(step 322), the advection and collision stages (steps 308-320) arerepeated. If the incremented timer indicates that the simulation iscomplete (step 322), results of the simulation are stored and/ordisplayed (step 324).

1. Boundary Conditions for Surface

To correctly simulate interactions with a surface, each facet must meetfour boundary conditions. First, the combined mass of particles receivedby a facet must equal the combined mass of particles transferred by thefacet (i.e., the net mass flux to the facet must equal zero). Second,the combined energy of particles received by a facet must equal thecombined energy of particles transferred by the facet (i.e., the netenergy flux to the facet must equal zero). These two conditions may besatisfied by requiring the net mass flux at each energy level (i.e.,energy levels one and two) to equal zero.

The other two boundary conditions are related to the net momentum ofparticles interacting with a facet. For a surface with no skin friction,referred to herein as a slip surface, the net tangential momentum fluxmust equal zero and the net normal momentum flux must equal the localpressure at the facet. Thus, the components of the combined received andtransferred momentums that are perpendicular to the normal n_(α) of thefacet (i.e., the tangential components) must be equal, while thedifference between the components of the combined received andtransferred momentums that are parallel to the normal n_(α) of the facet(i.e., the normal components) must equal the local pressure at thefacet. For non-slip surfaces, friction of the surface reduces thecombined tangential momentum of particles transferred by the facetrelative to the combined tangential momentum of particles received bythe facet by a factor that is related to the amount of friction.

2. Gather from Voxels to Facets

As a first step in simulating interaction between particles and asurface, particles are gathered from the voxels and provided to thefacets (step 308). As noted above, the flux of state i particles betweena voxel N(x) and a facet F_(α) is:Γ_(iα)(x)=N _(i)(x)V _(iα)(x).  Eq.(12)

From this, for each state i directed toward a facet F_(α)(c_(i)n_(α)<0), the number of particles provided to the facet F_(α) bythe voxels is:

$\begin{matrix}\begin{matrix}{\Gamma_{{i\;\alpha\; V}\rightarrow F} = {\sum\limits_{x}{\Gamma_{ia}(x)}}} \\{= {\sum\limits_{x}{{N_{i}(x)}{V_{i\;\alpha}(x)}}}}\end{matrix} & {{Eq}.\mspace{14mu}(13)}\end{matrix}$

Only voxels for which V_(iα) (x) has a non-zero value must be summed. Asnoted above, the size of the facets is selected so that V_(iα) (x) has anon-zero value for only a small number of voxels. Because V_(iα) (x) andP_(ƒ)(X) may have non-integer values, Γ_(α) (x) is stored and processedas a real number.

3. Move from Facet to Facet

Next, particles are moved between facets (step 310). If theparallelepiped G_(iα) for an incoming state (c_(i)n_(α)<0) of a facetF_(α) is intersected by another facet F_(β), then a portion of the statei particles received by the facet F_(α) will come from the facet F_(β).In particular, facet F_(α) will receive a portion of the state iparticles produced by facet F_(β) during the previous time increment.This relationship is illustrated in FIG. 10, where a portion 1000 of theparallelepiped G_(iα) that is intersected by facet F_(β) equals aportion 1005 of the parallelepiped G_(iβ) that is intersected by facetF_(α). As noted above, the intersected portion is denoted as V_(iα) (β).Using this term, the flux of state i particles between a facet F_(β) anda facet F_(α) may be described as:Γ_(iα)(β,t−1)=Γ_(i)(β)V _(iα)(β)/V _(iα)  Eq.(14)where Γ_(i)(β, t−1) is a measure of the state i particles produced bythe facet F_(β) during the previous time increment. From this, for eachstate i directed toward a facet F_(α) (c_(i)n_(α)<0), the number ofparticles provided to the facet F_(α) by the other facets is:

$\begin{matrix}\begin{matrix}{\Gamma_{{i\;\alpha\; F}\rightarrow F} = {\sum\limits_{\beta}{\Gamma_{ia}(\beta)}}} \\{= {\sum\limits_{\beta}{{\Gamma_{i}\left( {\beta,{t - 1}} \right)}{{V_{i\;\alpha}(\beta)}/V_{{i\;\alpha}\;}}}}}\end{matrix} & {{Eq}.\mspace{14mu}(15)}\end{matrix}$

and the total flux of state i particles into the facet is:

$\begin{matrix}\begin{matrix}{{\Gamma_{{i\; I\; N}\;}(\alpha)} = {\Gamma_{{i\;\alpha\; V}\rightarrow F} + \Gamma_{{i\;\alpha\; F}\rightarrow F}}} \\{= {{\sum\limits_{X}{{N_{i}(x)}{V_{i\;\alpha}(x)}}} + {\sum\limits_{\beta}{{\Gamma_{i}\left( {\beta,{t - 1}} \right)}{{V_{i\;\alpha}(\beta)}/V_{i\;\alpha}}}}}}\end{matrix} & {{Eq}.\mspace{14mu}(16)}\end{matrix}$

The state vector N(α) for the facet, also referred to as a facetdistribution function, has 54 entries corresponding to the 54 entries ofthe voxel state vectors. The input states of the facet distributionfunction N(α) are set equal to the flux of particles into those statesdivided by the volume V_(iα):N _(i)(α)=Γ_(iIN)(α)/V _(iα)  Eq.(17)for c_(i) n_(α)<0.

The facet distribution function is a simulation tool for generating theoutput flux from a facet, and is not necessarily representative ofactual particles. To generate an accurate output flux, values areassigned to the other states of the distribution function. Outwardstates are populated using the technique described above for populatingthe inward states:N _(i)(α)=Γ_(iOTHER)(α)/V  Eq.(18)for c_(i) n_(α)≧0, wherein Γ_(iOTHER)(α) is determined using thetechnique described above for generating Γ_(iIN)(α), but applying thetechnique to states (c_(i) n_(α)≧0) other than incoming states (c_(i)n_(α)<0)). In an alternative approach, Γ_(iOTHER)(α) may be generatedusing values of Γ_(iOUT)(α) from the previous time step so that:Γ_(iOTHER)(α,t)=Γ_(iOUT)(α,t−1).  Eq.(19)

For parallel states (c_(i)n_(α)=0), both V_(iα) and V_(iα)(x) are zero.In the expression for N_(i) (α), V_(iα) (x) appears in the numerator(from the expression for Γ_(iOTHER)(α) and V_(iα) appears in thedenominator (from the expression for N_(i)(α)). Accordingly, N_(i)(α)for parallel states is determined as the limit of N_(i)(α) as V_(iα) andV_(iα)(x) approach zero.

The values of states having zero velocity (i.e., rest states and states(0, 0, 0, 2) and (0, 0, 0, −2)) are initialized at the beginning of thesimulation based on initial conditions for temperature and pressure.These values are then adjusted over time.

4. Perform Facet Surface Dynamics

Next, surface dynamics are performed for each facet to satisfy the fourboundary conditions discussed above (step 312). A procedure forperforming surface dynamics for a facet is illustrated in FIG. 11.Initially, the combined momentum normal to the facet F_(α) is determined(step 1105) by determining the combined momentum P(α) of the particlesat the facet as:

$\begin{matrix}{{P(\alpha)} = {\sum\limits_{i}{c_{i}*N_{i}^{\alpha}}}} & {{Eq}.\mspace{14mu}(20)}\end{matrix}$for all i. From this, the normal momentum P_(n)(α) is determined as:P _(n)(α)=n _(α) ·P(α).  Eq.(21)

This normal momentum is then eliminated using a pushing/pullingtechnique (step 1110) to produce N_(n-)(α). According to this technique,particles are moved between states in a way that affects only normalmomentum. The pushing/pulling technique is described in U.S. Pat. No.5,594,671, which is incorporated by reference.

Thereafter, the particles of N_(n-)(α) are collided to produce aBoltzmann distribution N_(n-β)(α) (step 1115). As described below withrespect to performing fluid dynamics, a Boltzmann distribution may beachieved by applying a set of collision rules to N_(n-)(α).

An outgoing flux distribution for the facet F_(α) is then determined(step 1120) based on the incoming flux distribution and the Boltzmanndistribution. First, the difference between the incoming fluxdistribution Γ_(i)(α) and the Boltzmann distribution is determined as:ΔΓ_(i)(α)=Γ_(iIN)(α)−N _(n-βi)(α)V _(iα)  Eq.(22)

Using this difference, the outgoing flux distribution is:Γ_(iOUT)(α)=N _(n-βi)(α)V _(iα)−·Δ·Γ_(i)*(α),  Eq.(23)for n_(α)c_(i)>0 and where i* is the state having a direction oppositeto state i. For example, if state i is (1, 1, 0, 0), then state i* is(−1, −1, 0, 0). To account for skin friction and other factors, theoutgoing flux distribution may be further refined to:Γ_(iOUT)(α)N _(n-Bi)(α)V _(iα)−ΔΓ_(i*)(α)+C _(f)(n _(α) ·c _(i))[N_(n-Bi*)(α)−N _(n-Bi)(α)]V _(iα)+(n _(α) ·c _(i))(t _(1α) ·c _(i))ΔN_(j,1) V _(iα)+(n _(α) ·c _(i))(t _(2a) ·c _(i))ΔN _(j,2) V_(iα)  Eq.(24)for n_(α)c_(i)>0, where C_(ƒ) is a function of skin friction, t_(iα) isa first tangential vector that is perpendicular to n_(α), t_(2a), is asecond tangential vector that is perpendicular to both n_(α) and t_(1α),and ΔN_(j,1) and ΔN_(j,2) are distribution functions corresponding tothe energy (j) of the state i and the indicated tangential vector. Thedistribution functions are determined according to:

$\begin{matrix}{{\Delta\; N_{j,1,2}} = {{- \frac{1}{2\; j^{2}}}\left( {n_{\alpha} \cdot {\sum\limits_{i}{c_{i}c_{i}{{N_{n - {Bi}}(\alpha)} \cdot t_{1,{2\;\alpha}}}}}} \right)}} & {{Eq}.\mspace{14mu}(25)}\end{matrix}$where j equals 1 for energy level 1 states and 2 for energy level 2states.

The functions of each term of the equation for Γ_(iOUT) (α) are asfollows. The first and second terms enforce the normal momentum fluxboundary condition to the extent that collisions have been effective inproducing a Boltzmann distribution, but include a tangential momentumflux anomaly. The fourth and fifth terms correct for this anomaly, whichmay arise due to discreteness effects or non-Boltzmann structure due toinsufficient collisions. Finally, the third term adds a specified amountof skin fraction to enforce a desired change in tangential momentum fluxon the surface. Generation of the friction coefficient C_(ƒ) isdescribed below. Note that all terms involving vector manipulations aregeometric factors that may be calculated prior to beginning thesimulation.

From this, a tangential velocity is determined as:u _(i)(α)=(P(α)−P _(n)(α)n _(α))/ρ,  Eq.(26)where ρ is the density of the facet distribution:

$\begin{matrix}{\rho = {\sum\limits_{i}{N_{i}(\alpha)}}} & {{Eq}.\mspace{14mu}(27)}\end{matrix}$

As before, the difference between the incoming flux distribution and theBoltzmann distribution is determined as:ΔΓ_(i)(α)=Γ_(iIN)(α)−N _(n-βi)(α)V _(iα)  Eq.(28)

The outgoing flux distribution then becomes:Γ_(iOUT)(α)=N _(n-βi)(α)V _(iα)−ΔΓ_(i*)(α)+C _(ƒ)(n _(α) c _(i) {N_(n-βi)*(α)−N _(n-βi)(α)}V _(iα)  Eq.(29)which corresponds to the first two lines of the outgoing fluxdistribution determined by the previous technique but does not requirethe correction for anomalous tangential flux.

Using either approach, the resulting flux-distributions satisfy all ofthe momentum flux conditions, namely:

$\begin{matrix}{{\sum\limits_{i,{{c_{i} \cdot n_{\alpha}} > 0}}{c_{i}\Gamma_{i\;\alpha\;{OUT}^{-}}{\sum\limits_{i,{{c_{i} \cdot n_{\alpha}} < 0}}{c_{i}\Gamma_{i\;\alpha\; I\; N}}}}} = {{p_{\alpha}n_{\alpha}A_{\alpha}} - {C_{f}p_{\alpha}u_{\alpha}A_{\alpha}}}} & {{Eq}.\mspace{14mu}(30)}\end{matrix}$

where p_(α), is the equilibrium pressure at the facet F_(α) and is basedon the averaged density and temperature values of the voxels thatprovide particles to the facet, and u_(α), is the average velocity atthe facet.

To ensure that the mass and energy boundary conditions are met, thedifference between the input energy and the output energy is measuredfor each energy level j as:

$\begin{matrix}{{\Delta\;\Gamma_{\alpha\;{mj}}} = {{\sum\limits_{i,{{c_{ji} \cdot n_{\alpha}} < 0}}\Gamma_{\alpha\;{ji}\; I\; N}} - {\sum\limits_{i,{{c_{ji} \cdot n_{\alpha}} > 0}}\Gamma_{\alpha\;{ji}\;{OUT}}}}} & {{Eq}.\mspace{14mu}(31)}\end{matrix}$where the index j denotes the energy of the state i. This energydifference is then used to generate a difference term:

$\begin{matrix}{{\delta\;\Gamma_{\alpha\;{ji}}} = {V_{i\;\alpha}\Delta\;{\Gamma_{\alpha\;{mj}}/{\sum\limits_{i,{{c_{ji} \cdot n_{\alpha}} < 0}}V_{i\;\alpha}}}}} & {{Eq}.\mspace{14mu}(32)}\end{matrix}$

for c_(ji)n_(α)>0. This difference term is used to modify the outgoingflux so that the flux becomes:Γ_(αjiOUTƒ)=Γ_(αjiOUT)+δΓ_(αji)  Eq.(33)for c_(ji)n_(α)>0. This operation corrects the mass and energy fluxwhile leaving the tangential momentum flux unaltered. This adjustment issmall if the flow is approximately uniform in the neighborhood of thefacet and near equilibrium. The resulting normal momentum flux, afterthe adjustment, is slightly altered to a value that is the equilibriumpressure based on the neighborhood mean properties plus a correction dueto the non-uniformity or non-equilibrium properties of the neighborhood.

5. Move from Voxels to Voxels

Referring again to FIG. 3, particles are moved between voxels along thethree-dimensional rectilinear lattice (step 314). This voxel to voxelmovement is the only movement operation performed on voxels that do notinteract with the facets (i.e., voxels that are not located near asurface). In typical simulations, voxels that are not located nearenough to a surface to interact with the surface constitute a largemajority of the voxels.

Each of the separate states represents particles moving along thelattice with integer speeds in each of the three dimensions: x, y, andz. The integer speeds include: 0, ±1, and ±2. The sign of the speedindicates the direction in which a particle is moving along thecorresponding axis.

For voxels that do not interact with a surface, the move operation iscomputationally quite simple. The entire population of a state is movedfrom its current voxel to its destination voxel during every timeincrement. At the same time, the particles of the destination voxel aremoved from that voxel to their own destination voxels. For example, anenergy level 1 particle that is moving in the +1x and +1y direction (1,0, 0) is moved from its current voxel to one that is +1 over in the xdirection and 0 for other direction. The particle ends up at itsdestination voxel with the same state it had before the move (1,0,0).Interactions within the voxel will likely change the particle count forthat state based on local interactions with other particles andsurfaces. If not, the particle will continue to move along the latticeat the same speed and direction.

The move operation becomes slightly more complicated for voxels thatinteract with one or more surfaces. This can result in one or morefractional particles being transferred to a facet. Transfer of suchfractional particles to a facet results in fractional particlesremaining in the voxels. These fractional particles are transferred to avoxel occupied by the facet. For example, referring to FIG. 9, when aportion 900 of the state i particles for a voxel 905 is moved to a facet910 (step 308), the remaining portion 915 is moved to a voxel 920 inwhich the facet 910 is located and from which particles of state i aredirected to the facet 910. Thus, if the state population equaled 25 andV_(iα)(x) equaled 0.25 (i.e., a quarter of the voxel intersects theparallelepiped G_(iα)), then 6.25 particles would be moved to the facetF_(α) and 18.75 particles would be moved to the voxel occupied by thefacet F_(α). Because multiple facets could intersect a single voxel, thenumber of state i particles transferred to a voxel N(ƒ) occupied by oneor more facets is:

$\begin{matrix}{{N_{i}(f)} = {{N_{i}(x)}\left( {1 - {\sum\limits_{\alpha}{V_{i\;\alpha}(x)}}} \right)}} & {{Eq}.\mspace{14mu}(34)}\end{matrix}$where N(x) is the source voxel.

6. Scatter from Facets to Voxels

Next, the outgoing particles from each facet are scattered to the voxels(step 316). Essentially, this step is the reverse of the gather step bywhich particles were moved from the voxels to the facets. The number ofstate i particles that move from a facet F_(α), to a voxel N(x) is:

$\begin{matrix}{N_{{\alpha\;{iF}}\rightarrow V} = {\frac{1}{P_{f}(x)}{V_{\alpha\; i}(x)}{\Gamma_{\alpha\;{iOUT}_{f}}/V_{\alpha\; i}}}} & {{Eq}.\mspace{14mu}(35)}\end{matrix}$where P_(f)(x) accounts for the volume reduction of partial voxels. Fromthis, for each state i, the total number of particles directed from thefacets to a voxel N_((x)) is:

$\begin{matrix}{N_{{iF}\rightarrow V} = {\frac{1}{P_{f}(x)}{\sum\limits_{\alpha}{{V_{\alpha\; i}(x)}{\Gamma_{\alpha\;{iOUT}_{f}}/V_{\alpha\; i}}}}}} & {{Eq}.\mspace{14mu}(36)}\end{matrix}$

After scattering particles from the facets to the voxels, combining themwith particles that have advected in from surrounding voxels, andintegerizing the result, it is possible that certain directions incertain voxels may either underflow (become negative) or overflow(exceed 255 in an eight-bit implementation). This would result in eithera gain or loss in mass, momentum and energy after these quantities aretruncated to fit in the allowed range of values. To protect against suchoccurrences, the mass, momentum and energy that are out of bounds areaccumulated prior to truncation of the offending state. For the energyto which the state belongs, an amount of mass equal to the value gained(due to underflow) or lost (due to overflow) is added back to randomly(or sequentially) selected states having the same energy and that arenot themselves subject to overflow or underflow. The additional momentumresulting from this addition of mass and energy is accumulated and addedto the momentum from the truncation. By only adding mass to the sameenergy states, both mass and energy are corrected when the mass counterreaches zero. Finally, the momentum is corrected using pushing/pullingtechniques until the momentum accumulator is returned to zero.

7. Perform Fluid Dynamics

Finally, fluid dynamics are performed (step 318). This step may bereferred to as microdynamics or intravoxel operations. Similarly, theadvection procedure may be referred to as intervoxel operations. Themicrodynamics operations described below may also be used to collideparticles at a facet to produce a Boltzmann distribution.

The fluid dynamics is ensured in the lattice Boltzmann equation modelsby a particular collision operator known as the BGK collision model.This collision model mimics the dynamics of the distribution in a realfluid system. The collision process can be well described by theright-hand side of Equation 1 and Equation 2. After the advection step,the conserved quantities of a fluid system, specifically the density,momentum and the energy are obtained from the distribution functionusing Equation 3. From these quantities, the equilibrium distributionfunction, noted by ƒ^(eq) in equation (2), is fully specified byEquation (4). The choice of the velocity vector set c_(i), the weights,both are listed in Table 1, together with Equation 2 ensures that themacroscopic behavior obeys the correct hydrodynamic equation.

E. Variable Resolution

Referring to FIG. 12, variable resolution (as illustrated in FIGS. 6 and7 and discussed above) employs voxels of different sizes, hereinafterreferred to as coarse voxels 12000 and fine voxels 1205. (The followingdiscussion refers to voxels having two different sizes; it should beappreciated that the techniques described may be applied to three ormore different sizes of voxels to provide additional levels ofresolution.) The interface between regions of coarse and fine voxels isreferred to as a variable resolution (VR) interface 1210.

When variable resolution is employed at or near a surface, facets mayinteract with voxels on both sides of the VR interface. These facets areclassified as VR interface facets 1215 (F_(αIC)) or VR fine facets 1220(F_(αIF)). A VR interface facet 1215 is a facet positioned on the coarseside of the VR interface and having a coarse parallelepiped 1225extending into a fine voxel. (A coarse parallelepiped is one for whichc_(i) is dimensioned according to the dimensions of a coarse voxel,while a fine parallelepiped is one for which c_(i) is dimensionedaccording to the dimensions of a fine voxel.) A VR fine facet 1220 is afacet positioned on the fine side of the VR interface and having a fineparallelepiped 1230 extending into a coarse voxel. Processing related tointerface facets may also involve interactions with coarse facets 1235(F_(αC)) and fine facets 1240 (F_(αF)).

For both types of VR facets, surface dynamics are performed at the finescale, and operate as described above. However, VR facets differ fromother facets with respect to the way in which particles advect to andfrom the VR facets.

Interactions with VR facets are handled using a variable resolutionprocedure 1300 illustrated in FIG. 13. Most steps of this procedure arecarried out using the comparable steps discussed above for interactionswith non-VR facets. The procedure 1300 is performed during a coarse timestep (i.e., a time period corresponding to a coarse voxel) that includestwo phases that each correspond to a fine time step. The facet surfacedynamics are performed during each fine time step. For this reason, a VRinterface facet F_(αIC) is considered as two identically sized andoriented fine facets that are referred to, respectively, as a blackfacet F_(αICb) and a red facet F_(αICr). The black facet F_(αICb) isassociated with the first fine time step within a coarse time step whilethe red facet F_(αICr) associated with the second fine time step withina coarse time step.

Initially, particles are moved (advected) between facets by a firstsurface-to-surface advection stage (step 1302). Particles are moved fromblack facets F_(αICb) to coarse facets F_(βC) with a weighting factor ofV_(−αβ) that corresponds to the volume of the unblocked portion of thecoarse parallelepiped (FIG. 12, 1225) that extends from a facet F_(α)and that lies behind a facet F_(β) less the unblocked portion of thefine parallelepiped (FIG. 12, 1245) that extends from the facet F_(α)and that lies behind the facet F_(β). The magnitude of c_(i) for a finevoxel is one half the magnitude of c_(i) for a coarse voxel. Asdiscussed above, the volume of a parallelepiped for a facet F_(α) isdefined as:V _(iα) =|c _(i) n _(α) |A _(α)  Eq.(37)

Accordingly, because the surface area A_(α) of a facet does not changebetween coarse and fine parallelepipeds, and because the unit normaln_(α) always has a magnitude of one, the volume of a fine parallelepipedcorresponding to a facet is one half the volume of the correspondingcoarse parallelepiped for the facet.

Particles are moved from coarse facets F_(αC) to black facets F_(βICb)with a weighting factor of V_(αβ) that corresponds to the volume of theunblocked portion of the fine parallelepiped that extends from a facetF_(α) and that lies behind a facet F_(β).

Particles are moved from red facets F_(αICr) to coarse facets F_(αβ)with a weighting factor of V_(αβ), and from coarse facets F_(αC) to redfacets F_(βICr) with a weighting factor of V_(−αβ).

Particles are moved from red facets F_(αICr) to black facets F_(βICb)with a weighting factor of V_(αβ). In this stage, black-to-redadvections do not occur. In addition, because the black and red facetsrepresent consecutive time steps, black-to-black advections (orred-to-red advections) never occur. For similar reasons, particles inthis stage are moved from red facets F_(αICr) to fine facets F_(βIF) orF_(βF) with a weighting factor of V_(αβ), and from fine facets F_(αIF)or F_(αF) to black facets F_(αICb) with the same weighting factor.

Finally, particles are moved from fine facets F_(αIF) or F_(αF) to otherfine facets F_(βIF) or F_(βF) with the same weighting factor, and fromcoarse facets F_(αC) to other coarse facets F_(C) with a weightingfactor of V_(Cαβ) that corresponds to the volume of the unblockedportion of the coarse parallelepiped that extends from a facet F_(α) andthat lies behind a facet F_(β).

After particles are advected between surfaces, particles are gatheredfrom the voxels in a first gather stage (steps 1304-1310). Particles aregathered for fine facets F_(αF) from fine voxels using fineparallelepipeds (step 1304), and for coarse facets F_(αC) from coarsevoxels using coarse parallelepipeds (step 1306). Particles are thengathered for black facets F_(αIRb) and for VR fine facets F_(αIF) fromboth coarse and fine voxels using fine parallelepipeds (step 1308).Finally, particles are gathered for red facets F_(αIRr) from coarsevoxels using the differences between coarse parallelepipeds and fineparallelepipeds (step 1310).

Next, coarse voxels that interact with fine voxels or VR facets areexploded into a collection of fine voxels (step 1312). The states of acoarse voxel that will transmit particles to a fine voxel within asingle coarse time step are exploded. For example, the appropriatestates of a coarse voxel that is not intersected by a facet are explodedinto eight fine voxels oriented like the microblock of FIG. 4. Theappropriate states of coarse voxel that is intersected by one or morefacets are exploded into a collection of complete and/or partial finevoxels corresponding to the portion of the coarse voxel that is notintersected by any facets. The particle densities N_(i)(x) for a coarsevoxel and the fine voxels resulting from the explosion thereof areequal, but the fine voxels may have fractional factors P_(f) that differfrom the fractional factor of the coarse voxel and from the fractionalfactors of the other fine voxels.

Thereafter, surface dynamics are performed for the fine facets F_(αIF)and F_(αF) (step 1314), and for the black facets F_(αICb) step 1316).Dynamics are performed using the procedure illustrated in FIG. 11 anddiscussed above.

Next, particles are moved between fine voxels (step 1318) includingactual fine voxels and fine voxels resulting from the explosion ofcoarse voxels. Once the particles have been moved, particles arescattered from the fine facets F_(αIF) and F_(αF) to the fine voxels(step 1320).

Particles are also scattered from the black facets F_(αICb) to the finevoxels (including the fine voxels that result from exploding a coarsevoxel) (step 1322). Particles are scattered to a fine voxel if the voxelwould have received particles at that time absent the presence of asurface. In particular, particles are scattered to a voxel N(x) when thevoxel is an actual fine voxel (as opposed to a fine voxel resulting fromthe explosion of a coarse voxel), when a voxel N(x+c_(i)) that is onevelocity unit beyond the voxel N(x) is an actual fine voxel, or when thevoxel N(x+c_(i)) that is one velocity unit beyond the voxel N(x) is afine voxel resulting from the explosion of a coarse voxel.

Finally, the first fine time step is completed by performing fluiddynamics on the fine voxels (step 1324). The voxels for which fluiddynamics are performed do not include the fine voxels that result fromexploding a coarse voxel (step 1312).

The procedure 1300 implements similar steps during the second fine timestep. Initially, particles are moved between surfaces in a secondsurface-to-surface advection stage (step 1326). Particles are advectedfrom black facets to red facets, from black facets to fine facets, fromfine facets to red facets, and from fine facets to fine facets.

After particles are advected between surfaces, particles are gatheredfrom the voxels in a second gather stage (steps 1328-1330). Particlesare gathered for red facets F_(αIRr) from fine voxels using fineparallelepipeds (step 1328). Particles also are gathered for fine facetsF_(αF) and F_(αIF) from fine voxels using fine parallelepipeds (step1330).

Thereafter, surface dynamics are performed for the fine facets F_(αIF)and F_(αF) (step 1332), for the coarse facets F_(αC) (step 1134), andfor the red facets F_(αICr) (step 1336) as discussed above.

Next, particles are moved between voxels using fine resolution (step1338) so that particles are moved to and from fine voxels and finevoxels representative of coarse voxels. Particles are then moved betweenvoxels using coarse resolution (step 1340) so that particles are movedto and from coarse voxels.

Next, in a combined step, particles are scattered from the facets to thevoxels while the fine voxels that represent coarse voxels (i.e., thefine voxels resulting from exploding coarse voxels) are coalesced intocoarse voxels (step 1342). In this combined step, particles arescattered from coarse facets to coarse voxels using coarseparallelepipeds, from fine facets to fine voxels using fineparallelepipeds, from red facets to fine or coarse voxels using fineparallelepipeds, and from black facets to coarse voxels using thedifferences between coarse parallelepipeds and find parallelepipeds.Finally, fluid dynamics are performed for the fine voxels and the coarsevoxels (step 1344).

F. Porous Media Interface Model

The resistance of fluid flow through a porous media (PM) is commonlydescribed by Darcy's law, which states that the pressure drop betweentwo points is proportional to the flow rate “ρu” and the distance Lbetween the two points:p2−p1=σLρu,where “σ” is the PM resistivity. For flow through a PM with highporosity Φ close to 1, where porosity (between 0 and 1) is defined asthe volume ratio of PM pores, the flow details at the interface betweenthe PM and fluid generally may be neglected. However, for a PM with lowporosity, the interface effect may be significant for certain types ofapplications, such as flow acoustics.

For example, FIG. 14 illustrates a fluid F flowing toward an interfacesurface 1401 of a porous medium PM with porosity Φ. In a unit area 1402on the interface surface 1401, the fraction of the surface that ispenetrable and into which the fluid may flow is only Φ. By contrast, thefraction of the surface that is blocked by the PM solid structure is1−Φ. As a result, only a Φ fraction of fluid F in each unit area on thesurface can enter the PM, while the 1−Φ fractional part of the fluid Fis blocked by the PM solid wall and stays on the fluid side of theinterface surface 1401. In a kinetic approach, such as the LBM approach,such a partial fluid penetration can be realized efficiently. Since afluid is represented by fluid particles, i.e., fluid fluxes, such asmass momentum energy fluxes, and particle distributions, the Φ fractionof particles is allowed to move into the PM during particle advection,and the 1−Φ fraction of particles is constrained by the PM solid wallboundary condition (BC). Here, the fluid particles may include particledistributions or fluxes of hydrodynamic and thermodynamic propertiessuch as mass fluxes, momentum fluxes, and energy fluxes. Additionally,the fluid particles, or elements, may include properties such as mass,density, momentum, pressure, velocity, temperature, and energy.Moreover, the elements may be associated with any fluid, flow, orthermodynamic related quantity although not exhaustively identifiedherein.

Either a frictional wall (bounce-back or turbulent wall) BC or africtionless wall BC can be applied. The fraction of particles allowedto move into the PM affects the mass and momentum conditions in thedirection normal to the interface. For the tangential behavior at theinterface, either a frictionless wall or a frictional wall BC can beapplied (as is true for a “typical” wall boundary). A frictionless wallBC maintains the surface tangential fluid velocity on the wall by notmodifying the flux of tangential momentum at the interface. A frictionalwall BC does alter the tangential momentum flux to achieve, for example,a no-slip wall boundary condition, or a turbulent wall model. These wallBCs ensure that there is zero mass flux across the wall. When porosity Φequals 1, the wall portion of the PM effectively disappears and thepartial wall model ceases to have effect.

While the fraction of particles passing from the fluid into the PM iscontrolled by the porosity of the PM, particles leaving the PM can movefreely because they encounter no solid obstacle from the fluid side.These particles together with the particles that were blocked fromentering the PM form the total particle flow into the fluid side.

The PM interface X can be described by so-called double-sided surfaceelements (i.e, surfels), as shown in FIG. 15. In such double-sidedsurfels, a set of paired surfels S form a double-layered surface havingan inner surface A and outer surface B. The inner surface A interactswith the PM and the outer surface B interacts with fluid domain Fd.There is no gap between the inner and outer surfaces A and B. Forconvenience of computation, each inner surfel has the exact same shapeand size as its paired outer surfel, and each inner surfel is only intouch with the paired outer surfel. The standard surfel gather andscatter scheme is performed on each side of the surface A, B, and withthe condition that the Φ_(f) fraction of incoming particles from thefluid side F pass through to the PM side while all of the incomingparticles Φ_(PM) from the PM side pass through to the fluid side F.Advantages of this approach include simplified handling of the complexPM interface, exact satisfaction of conservation laws, and easyrealization of specified fluid boundary conditions on PM interface.

This approach, in effect, introduces a PM interface resistance which isnot proportional to a PM thickness and therefore cannot be included inapproximation of Darcy's law. The approach accounts for the flow detailsat the PM interface and improves simulation results of certain types offlow problems, such as the modeling of acoustic absorption.

For example, referring to FIG. 16, in a system 1600 for modelingacoustic absorption, a fluid flow region FF may be adjacent to a regionPM occupied by a PM material with sound absorbing properties, with a PMinterface X providing the interface between the fluid flow region FF andthe region PM, and a wall interface Y providing the interface betweenthe region PM and a wall W. The fluid flow region FF and the region PMcan be, in effect, treated as two separate simulation spaces havingdifferent properties (e.g., in the region PM, an increased impedance maybe used to account for the presence of the PM impedance), with movementsΦ and 1−Φ between the two simulation spaces FF, PM being governed by theproperties of the PM interface, as discussed above.

Acoustic absorption by porous materials is an important topic inacoustics engineering. At a microscopic scale, the propagation of soundin porous media is difficult to characterize because of the topologicalcomplexity of the materials. At a macroscopic scale, porous materialswith high porosity can be treated as regions of fluid which havemodified properties relative to air. Sound propagation in such media canbe expressed in the form of two intrinsic, frequency-dependent, andvolumetric properties of the material: the characteristic impedance andthe complex acoustic wave number. Under certain assumptions, a givenvolumetric model for sound propagation in an absorbing material can beput in the form of locally-reacting, frequency-dependent, compleximpedance at the interface between two different media. For example,impedance models, such as Boundary Element Methods (BEM), FiniteElements Methods (FEM), and Statistical Energy Analysis (SEA) methods,and are implemented as boundary conditions in the frequency domain.

For problems involving flow-induced noise, suitable Computational FluidDynamics (CFD) and/or Computational AeroAcoustics (CAA) numericalmethods are non-linear and often time-explicit. For a time-explicitsolution, time-domain surface impedance boundary conditions couldlikewise allow modeling of acoustic absorption due to porous materials.However, even when a time-domain surface impedance formulation can bederived, stability and robustness appear to be challenging problems toovercome. An exemplary approach includes modeling of absorbing materialsas volumetric fluid regions, such that sound waves travel through thematerial and dissipate via a momentum sink. This is related to themethod for macroscopic modeling of flow through porous media achieved byrelating the momentum sink to the flow resistance of the materialfollowing Darcy's law. For an exemplary acoustic absorption modelingmethod, there is the question of how to determine the momentum sink toachieve a desired absorption behavior. If the acoustic absorption isgoverned (or at least dominated) by the same physical mechanisms as theflow resistivity, then the same momentum sink behavior used to achievethe correct flow resistivity for a particular porous material shouldalso achieve the correct acoustic absorption for that material. Thisapproach should be valid for any passive and homogeneous porousmaterial. Moreover, numerical stability problems should be removed sincethe impedance is realized in a way that is inherently well-posed (i.e.passive, causal, and real conditions are satisfied).

According to this exemplary approach, a time-explicit CFD/CAA solutionmethod based on the Lattice Boltzmann Method (LBM), which has evolvedover the last two decades as an alternative numerical method totraditional CFD, may be used. Unlike methods based on discretizing themacroscopic continuum equations, LBM starts from a “mesoscopic”Boltzmann kinetic equation to predict macroscopic fluid dynamics. Theresulting compressible and unsteady solution method may be used forpredicting a variety of complex flow physics, such as aeroacoustics andpure acoustics problems. A porous media model is used to represent theflow resistivity of various components, such as air filters, radiators,heat exchangers, evaporators, and other components, which areencountered in simulating flow, such as through HVAC systems, vehicleengine compartments, and other applications.

The propagation of sound waves inside a homogeneous and passiveabsorbing material with a porosity close to Φ=1 is macroscopically fullycharacterized by the material's characteristic impedance Z_(c)(ω) andcomplex wave number k(ω). By performing measurements on various porousand fibrous materials, many semi-empirical models are derived, such asthe Delany-Bazley or Allard-Champoux 3-parameter models. For example,the Allard-Champoux 3-parameter model is given by:

$\begin{matrix}{{Z_{c}(\omega)} = {\rho_{0}{c_{0}\left\lbrack {1 + {0.008127\; X^{- 0.75}} - {{j0}{.001228}\; X^{- 0.73}}} \right\rbrack}}} & \left( {{Eq}.\mspace{14mu} 38} \right) \\{{k(\omega)} = {\frac{\omega}{c_{0}}\left\lbrack {1 + {0.085787\; X^{- 0.70}} - {j\; 0.174919\; X^{- 0.59}}} \right\rbrack}} & \left( {{Eq}.\mspace{14mu} 39} \right)\end{matrix}$where ρ₀ is the density of air, c₀ the sound speed in air, and X is adimensionless parameter equal to X=ρ0ω/2πσ with σ the flow resistivity.This model is considered valid for 0.01<X<0.1. For the situation of alayer of porous material PM of uniform thickness “d” backed by animpervious rigid wall 1701, as shown in FIG. 17A, the complex impedanceZ_(s)(ω) at normal incidence at the air/material interface I_(am) is:

$\begin{matrix}{{Z_{s}(\omega)} = {{- j}\frac{1}{\phi}{Z_{c}(\omega)}{\cos\left( {{k(\omega)} \cdot d} \right)}}} & \left( {{Eq}.\mspace{14mu} 40} \right)\end{matrix}$As shown in FIG. 17B, in which there is a layer of air of thickness “e”in between the porous material PM and the rigid wall 1701, theexpression for Z_(s)(ω) becomes:

$\begin{matrix}{{Z_{s}(\omega)} = {{- \frac{1}{\phi}}{Z_{c}(\omega)}\frac{{{- j}\;{Z_{a}(\omega)}{\cot\left( {{k(\omega)} \cdot e} \right)}} + {Z_{c}(\omega)}}{{Z_{a}(\omega)} - {j\;{Z_{c}(\omega)}{\cot\left( {{k(\omega)} \cdot e} \right)}}}}} & \left( {{Eq}.\mspace{14mu} 41} \right)\end{matrix}$with Z_(a) (ω)=−jZ₀·cot(k₀(ω)·d, k₀(ω)=ω₀.

The complex surface impedance Z_(s)(ω) is expressed as a function of itsreal and imaginary parts, the resistance R(ω) and the reactance X(ω),respectively. For passive materials, characterized by R(ω)>0 (i.e.positive resistance), the material absorption coefficient α(ω) isdefined by:

$\begin{matrix}{{\alpha(\omega)} = {1 - {\frac{1 - Z_{s}}{1 + Z_{s}}}^{2}}} & \left( {{Eq}.\mspace{14mu} 42} \right)\end{matrix}$The surface impedance can be measured in an impedance tube using atwo-microphone method as described below.

The LBM-based method can be used to compute unsteady flow and thegeneration and propagation of acoustics waves. In LBM, external forcescan be included in the fluid dynamics by altering thelocal-instantaneous particle distributions during the collision step.The external force applied per unit time effectively becomes a momentumsource/sink. This technique can be used, for example, to model buoyancyeffects due to gravity. The method implements a porous media model byapplying an external force based on Darcy's law for flow resistivity asa function of flow velocity. The effect of a porous medium on the flowis achieved by removing an amount of momentum at each volumetriclocation of the porous region such that the correct pressure gradient isachieved, resulting in the correct overall pressure drop.

To assess the effect of the porous media model on acoustics propagation,a 3D circular impedance tube 1801, as shown in FIG. 18, can besimulated. For example, the tube dimensions are length L=0.772 m anddiameter D=0.0515 m, and the frequency range for valid usage of the tube1801 may be, for example, 100 Hz-3000 Hz. The tube walls 1802 arepresumed as rigid and frictionless, and a time-varying pressure boundarycondition representing white noise is applied at the inlet 1803. Thelayer of thickness “d” represents an absorbing material and is a porousmedia region PM, characterized by flow resistivity σ_(x) in thex-direction and infinite resistance in the other directions. An airlayer of thickness “e” can be included between the porous media regionPM and the right-hand side rigid wall 1804. Uniform grid resolution maybe used, for example, with 30 points per wavelength at ƒ=3000 Hz, i.e.Δx=1.7 mm, ensuring low numerical dissipation of acoustic waves. Thetime step is Δt=4.56×10⁻⁶ s and the simulations are run for a time ofT=2 s, which was observed to be a sufficient physical time for theresults to evolve beyond the startup transient period and providemeaningful statistics.

Pressure time histories are recorded inside the tube at two virtualmicrophones p₁(x₁, t) and p₂(x₂, t). Using x₁−x₂=s and x₁=l, the surfacecomplex impedance at x=0 is given by the following expression:

$\begin{matrix}{\frac{Z_{s}}{Z_{0}} = {j\frac{{h_{12}{\sin({kl})}} - {\sin\left\lbrack {k\left( {l - s} \right)} \right\rbrack}}{{\cos\left\lbrack {k\left( {l - s} \right)} \right\rbrack} - {h_{12}{\cos({kl})}}}}} & \left( {{Eq}.\mspace{14mu} 43} \right)\end{matrix}$with h₁₂ the complex transfer function between p₁ and p2, and wavenumber k=ω/c₀=2πƒ/c₀. From this expression, acoustic resistance,reactance, and absorption coefficient can be derived and compared tosemi-empirical models and experimental results.

TABLE 1 σ_(x) Equivalent Configuration d (mm) e (mm) (rayls/m) materialtype A 0 0 0 None (Air) B 26.5 0 23150 Felt C 26.5 48.5 23150 Felt D26.5 120.0 23150 Felt E 26.5 0.0 14150 Foam F 26.5 120.0 14150 Foam G26.5 0 200 Air filter

Simulated configurations, as shown in Table 1, are compared to theAllard-Champoux model (Equations 38 and 39). As shown by Configuration A(no absorbing layer), the convergence of the results with respect to thegrid resolution and characterization of any residual absorption of thenumerical system result. As shown in FIG. 19 for Configuration A,residual absorption is close to zero at low frequencies, while at highfrequencies (f>1500 Hz), some absorption is measured for the coarserresolutions. This is related to increasing numerical dissipation ofacoustics in both the air and at the solid boundaries as the grid ismade coarser. For 40 points per wavelength (ppw), the residualabsorption is less than 5% for f<3000 Hz, which may be considered to besatisfactory. The peaks at 1900 Hz and 2650 Hz are related to poles ofEquation 42, and correspond to numerical and signal processingartifacts. As shown by Configuration G, which corresponds to a typicalair filter in an automotive HVAC system, with a flow resistivity σx=100rayls/m.

In FIGS. 20-22, some exemplary preliminary results are shown with a 30ppw simulation with LBM-PM model results, Allard-Champoux model results,and experimental data. In FIG. 20 for Configuration B, the thickness “d”of the PM is 26.5 mm, the thickness “e” of the air is 0.0 mm, and flowresistivity “σ” is 23150 rayls/m. In FIG. 21 for Configuration C, thethickness “d” of the PM is 26.5 mm, the thickness “e” of the air is 48.5mm, and the flow resistivity “a” is 23150 rayls/m. In FIG. 22 forConfiguration D, the thickness “d” of the PM is 120.0 mm, the thickness“e” of the air is 48.5 mm, and the flow resistivity “σ” is 23150rayls/m. Here, as demonstrated by FIGS. 20-22, the validity of theAllard-Champoux model is confirmed, and the simulation results alsocorrelate well to the LBM-PM model and to the experimental results. Thefrequency dependence of the absorption coefficient is well-captured foreach Configuration, including non-monotonic behavior.

For configuration G, corresponding to a typical air filter in a HVACsystem, for example, the acoustic absorption is relatively small. Thus,the LBM-PM model approach correctly captures both flow and acousticeffects, even for a material that has a significant flow resistanceeffect but a negligible effect on acoustics.

Accurate prediction of fan noise is an important issue in the field ofaeroacoustics. As vehicle manufacturers seek to reduce the noise levelsexperienced by passengers, the noise due to the heating, ventilation,and air conditioning (HVAC) system becomes a target for improvedacoustic performance. The HVAC system is complex, consisting of a blowerand mixing unit coupled to many ducts through which air is transportedto various locations, including faces and feet of front and rearpassengers, as well as windshield and sideglass defrost. The blower mustsupply sufficient pressure head to achieve desired air flow rates foreach thermal comfort setting. Noise is generated due to the blowerrotation, and by the turbulent air flow in the mixing unit through thetwists and turns of the ducts, and exiting the registers (ventilationoutlets). When designing an HVAC system it is difficult to predictwhether noise targets will be met, and to find the best compromisebetween flow, thermal, and acoustic performance while meeting packagingconstraints. The effects of integrating the HVAC system into thevehicle, which changes the performance relative to the test bench, mustalso be accounted for.

For example, as exemplified in FIG. 23, noise heard by passengers due tothe HVAC system of a vehicle 2300, which involves many sources andpaths, may be the noise absorbed at an interior cabin 2302. For example,the noise may result from a blower of the vehicle that includes a radialfan which generates noise from the interaction of the moving blades withthe surrounding air, and the impact of the moving air on nearby staticcomponents, such as seats 2304 and interior roof 2306 of the vehicle.This fan noise is acoustically propagated through the complex network ofducts, out of the registers, and into the interior cabin 2302. The ductand mixing unit flow noise sources are generated mainly by flowseparations and vortices resulting from the detailed geometric features,and are also acoustically propagated through the system. Noise due tothe flow exiting the registers depends on the fine details of the grilland its orientation, and the resulting outlet jets which mix with theambient air and may impact surfaces, such as the windshield 2308 (e.g.,for defrost). Therefore, the requirements for numerical flow-acousticpredictions may be accomplished using the exemplary modeling, asdetailed above, whereby the interior cabin 2302 may be considered afluid of a first volume and the static components and surfaces withinthe interior cabin 2302 may be considered a second volume occupied by aporous medium. By implementing the exemplary modeling, complexgeometries may be investigated to provide predictions of the fan andflow induced noise sources, and their acoustic propagation all the waythrough the system to the locations of the passengers at the interiorcabin 2302 of the vehicle 2300.

The exemplary modeling provides accurate numerical noise prediction forfully detailed automotive HVAC systems, such as accurate predictions ofthe complex flow structures, corresponding noise sources, and resultingpropagated acoustics to the passenger head space locations, includingeffects of geometric details throughout the integrated system. Thetransient flow characteristics and acoustics can be determined,including the rotating fan flow and noise, as well as direct predictionof acoustic propagation throughout the system. The exemplary modelingcan obtain early noise assessment of proposed designs and evaluatepotential design options, and/or diagnose and improve noise problems onan existing design. In addition, the exemplary model providesvisualization capabilities to allow identification and insight intosources of noise, including band-filtered pressure analyses to isolatephenomena at specific frequency bands of interest. Predicted spectra atpassenger locations can be converted to audio files for comparativelistening to the effects of various design options. The exemplarymodeling also provides accurate HVAC system pressures, flow rates, andthermal mixing behavior—hence it can be used to assessmulti-disciplinary design tradeoffs to design the HVAC system withoptimal aero, thermal, and acoustic performance.

In another example, the operation of transportation vehicles and heavymachinery results in sound propagated through the air which reachespeople in the surrounding areas and is known as community (orenvironmental) noise. Increased usage of air and ground transportationhas brought significant increases in community noise, with provenadverse health effects. This noise pollution is now considered a seriousproblem and is government regulated in most countries, with the specificregulations varying by industry and vehicle type as well as from countryto country. It is important to design products that do not exceedregulated noise targets, which involve the sound levels reaching anobserver at a specified location or distance relative to the movingvehicle or stationary equipment. To assess whether a target will be met,key sources of noise generated by turbulent flow or mechanical vibrationin the near-field and the resulting sound propagation to the observer inthe far-field must be determined.

A major part of designing towards meeting noise targets is to assess andreduce noise sources while dealing with the multitude of other designconstraints. Experimental testing challenges also include wind tunnelspace limitations for extending measurements to the far-field, andrelating stationary source wind tunnel measurements to the real lifemoving source scenario. A key challenge faced by both experimental andnumerical techniques in the identification of flow-induced noise sourcesis that sound propagated to the far-field consists of pressureperturbations, which may be very small relative to the turbulentpressure fluctuations in the near-field source region. Hence, accordingto the exemplary modeling, as detailed above, predictions of both thenoise sources and the resulting acoustic propagation may be accomplishedto achieve highly accurate transient flow behavior, and sufficiently lowdissipation and dispersion, to resolve small amplitude fluctuations overthe frequency range of interest. Moreover, in typical applications, suchas aircraft or train certification, the far-field noise target involveslarge distances making it impractical to extend the computational domainto include both source region and the observer.

To predict far-field noise, the exemplary modeling may be used toprovide detailed flow behavior and resulting near-field sources foreither a vehicle component of interest, such as an aircraft landing gearassembly 2400, as shown in FIG. 24, or a complete vehicle. According tothe exemplary modeling, transient solutions accurately predict thecomplex time-dependent flow structures, corresponding noise sources, andcan accommodate the required realistic detailed geometry, such variouscomponents 2402 of the aircraft landing gear assembly 2400. The resultscan be coupled to a far-field propagation module to easily andefficiently predict the far-field noise at any location, whereby theregion surrounding the aircraft landing gear assembly 2400 may beconsidered a fluid of a first volume and regions between the components2402 of the aircraft landing gear assembly 2400 may be represented as asecond volume occupied by a porous medium. The exemplary modeling allowsfor early noise assessment and optimization, including noisecertification evaluation (e.g. using the Evolution of Perceived NoiseLevel EPNL metric) before a final prototype is built. In addition,visualization of the exemplary modeling can provide insight into sourcesof noise, including band-filtered pressure analyses to isolate phenomenaat specific frequency bands of interest, for example to find the causeof a peak observed in a far-field spectrum.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made without departingfrom the spirit and scope of the claims. Accordingly, otherimplementations are within the scope of the following claims.

What is claimed is:
 1. A computer-implemented method for determiningsound wave movement through at least one porous material in or on aphysical or mechanical device, the method comprising: modeling the atleast one porous material as a first volumetric fluid region, in asimulation space, in which fluid flows and sound waves travel throughthe at least one porous material and dissipate via a momentum sink thataccounts for pressure and acoustic losses of the at least one porousmaterial modeled in the simulation space; wherein the momentum sink thatis modeled removes one or more amounts of momentum at a plurality ofvolumetric locations within the first volumetric fluid regionrepresenting the at least one porous material to model acoustical lossesin the at least one porous material; simulating, in a simulation space,fluid flow and propagation of sound waves in a first volumetric fluidregion representing the at least one porous material, the activity ofthe fluid being simulated so as to model movement of elements within thefirst volumetric fluid region and using a first model having a first setof parameters that comprises a parameter for the momentum sink;obtaining, based on using the momentum sink in simulating fluid flow andpropagation of sound waves in the first volumetric fluid regionrepresenting the at least one porous material, (i) pressure losses and(ii) acoustical losses in the at least one porous material; simulating,in the simulation space, fluid flow and propagation of sound waves in asecond volumetric fluid region representing a portion of the physical ormechanical device, the second volumetric fluid region adjoining thefirst volumetric fluid region representing the at least one porousmaterial in the simulation space, the activity in the second volumetricfluid region being simulated so as to model movement of elements withinthe second volumetric fluid region and using a second model having asecond set of parameters; and modeling, based on the simulations, fluidflow and propagation of sound waves in (i) the first volumetric fluidregion representing the at least one porous material, (ii) the secondvolumetric fluid region representing the portion of physical ormechanical device, and (iii) between the first and second volumetricfluid regions in the simulation space to model fluid flow andpropagation of sounds waves in the system.
 2. The method of claim 1,further comprising storing, in a computer-accessible memory, a first setof state vectors for voxels of the first volumetric fluid regionrepresenting the at least one porous material and a second set of statevectors for voxels of the second volumetric fluid region representingthe physical or mechanical device, wherein each of the state vectorscomprises a plurality of entries that correspond to particular momentumstates of possible momentum states at a corresponding voxel, wherein:simultaneously simulating activity of the fluid in the first volumetricfluid region representing the at least one porous material comprises:performing first interaction operations on the first set of statevectors, the first interaction operations modeling interactions betweenelements of different momentum states according to the first model, andperforming first move operations on the first set of state vectors toreflect movement of elements to new voxels in the first volume accordingto the first model; and simulating fluid flow and propagation ofacoustic wave in the second volume comprises: performing secondinteraction operations on the second set of state vectors, the secondinteraction operations modeling interactions between elements ofdifferent momentum states according to the second model, and performingsecond move operations on the second set of state vectors to reflectmovement of elements to new voxels in the second volume according to thesecond model.
 3. The method of claim 2, further comprising simulatingacoustic absorption between the first volumetric fluid regionrepresenting the at least one porous material and the second volumetricfluid region representing the physical or mechanical device at aninterface between the first volumetric fluid region representing the atleast one porous material and the second volumetric fluid region bysimulating movement of elements between state vectors of the first setof state vectors for voxels in the first volumetric fluid region at theinterface and state vectors of the second set of state vectors forvoxels in the second volumetric fluid region at the interface, andwherein movement of elements from the first volumetric fluid region tothe second volumetric fluid region is governed by a first set ofconstraints and movement of elements from the second volume to the firstvolume is governed by a second set of constraints.
 4. The method ofclaim 3, wherein the first set of constraints permits a fraction ofelements oriented to move from the first volumetric fluid regionrepresenting the at least one porous material to the second volumetricfluid region representing the physical or mechanical device to actuallymove from the first volumetric fluid region to the second volumetricfluid region, the fraction corresponding to a porosity of the porousmaterial being simulated.
 5. The method of claim 4, wherein the secondset of constraints permits all elements oriented to move from the secondvolumetric fluid region representing the physical or mechanical deviceto the first volumetric fluid region representing the at least oneporous material to actually move from the second volumetric fluid regionto the first volumetric fluid region.
 6. The method of claim 2, whereinthe second set of parameters of the second model differs from the firstset of parameters of the first model in a way that accounts for aporosity of the porous material being simulated.
 7. The method of claim2, wherein the second set of parameters of the second model differs fromthe first set of parameters of the first model in a way that accountsfor one or more of acoustic resistance, acoustic absorption, andacoustic impedance of the porous material being simulated.
 8. The methodof claim 2, wherein the second set of parameters of the second modeldiffers from the first set of parameters of the first model in a waythat accounts for one or more of tortuosity, characteristic viscouslength, thermal characteristic length, and thermal permeability of theporous material being simulated.
 9. The method of claim 1, whereinsimulating movement of elements between the first volumetric fluidregion representing the at least one porous material and the secondvolumetric fluid region representing the physical or mechanical deviceat the interface between the first volumetric fluid region and thesecond volumetric fluid region comprises simulating movement of elementsbetween a region of the first volume at the interface and a region ofthe second volumetric fluid region at the interface, and whereinmovement of elements from the first volumetric fluid region to thesecond volumetric fluid is governed by a first set of constraints andmovement of elements from the second volumetric fluid region to thefirst volumetric fluid region is governed by a second set of constraintsthat differs from the first set of constraints.
 10. The method of claim9, wherein the first set of constraints permits a fraction of elementsoriented to move from the first volumetric fluid region representing theat least one porous material to the second volumetric fluid regionrepresenting the physical or mechanical device to actually move from thefirst volume to the second volumetric fluid region, the fractioncorresponding to a porosity of the porous material being simulated. 11.The method of claim 1, wherein the second set of constraints permits allelements oriented to move from the second volumetric fluid regionrepresenting the physical or mechanical device to the first volumetricfluid region representing the at least one porous material to actuallymove from the second volumetric fluid region to the first volumetricfluid region.
 12. The method of claim 1, wherein the second set ofparameters of the second model differs from the first set of parametersof the first model in a way that accounts for a porosity of the porousmaterial being simulated.
 13. The method of claim 1, wherein the secondset of parameters of the second model differs from the first set ofparameters of the first model in a way that accounts for one or more ofacoustic resistance, acoustic absorption, and acoustic impedance of theporous material being simulated.
 14. The method of claim 1, wherein thesecond set of parameters of the second model differs from the first setof parameters of the first model in a way that accounts for one or moreof tortuosity, characteristic viscous length, thermal characteristiclength, and thermal permeability of the porous material being simulated.15. The method of claim 1, wherein the elements comprise particledistributions.
 16. The method of claim 1, wherein the elements compriseone or more of mass, density, momentum, pressure, velocity, temperature,energy, mass fluxes, momentum fluxes, and energy fluxes within thefluid.
 17. The method of claim 1, the first volumetric fluid regionrepresenting the at least one porous material represents an interiorcabin of a vehicle.
 18. The method of claim 1, wherein the secondvolumetric fluid region representing the physical or mechanical deviceincludes static components and surfaces within the interior cabin of thevehicle.
 19. A computer-implemented method determining sound wavemovement through at least one porous material in or on a physical ormechanical device, the method comprising: modeling the at least oneporous material as a first volumetric fluid region representing the atleast one porous material, in a simulation space, in which fluid flowsand sound waves travel through the at least one porous material anddissipate via a momentum sink that accounts for pressure and acousticlosses of the at least one porous material modeled in the simulationspace; wherein the momentum sink that is modeled removes one or moreamounts of momentum at a plurality of volumetric locations within thefirst volumetric fluid region representing the at least one porousmaterial to model and acoustical losses in the at least one porousmaterial; simultaneously simulating, in a simulation space, fluid flowand propagation of acoustic waves in a first volumetric fluid regionrepresenting the at least one porous material, the activity of the fluidin the first volume being simulated so as to model movement of elementswithin the first volumetric fluid region and using a first model havinga first set of parameters that comprises a parameter for the momentumsink; obtaining, based on using the momentum sink in simulating fluidflow and propagation of acoustic waves in the first volumetric fluidregion representing the at least one porous material, (i) pressurelosses and (ii) acoustical losses of the at least one porous materialmodeled in the first volumetric fluid region; simulating, in thesimulation space, fluid flow and propagation of acoustic waves in asecond volumetric fluid region representing a portion of the physical ormechanical device, the second volumetric fluid region adjoining thefirst volumetric fluid region representing the at least one porousmaterial in the simulation space, the activity in the second volumetricfluid region being simulated so as to model movement of elements withinthe second volumetric fluid region and using a second model having asecond set of parameters; and simulating flow resistance and acousticabsorption due to movement of elements between the first volumetricfluid region representing the at least one porous material and thesecond volumetric fluid region representing the portion of physical ormechanical device at an interface between the first volumetric fluidregion representing the at least one porous material and the secondvolumetric fluid region by simulating movement of elements between aregion of the first volumetric fluid region at the interface and aregion of the second volumetric fluid region at the interface with themovement of elements from the first volumetric fluid region to thesecond volumetric fluid region being governed by a first set ofconstraints and movement of elements from the second volumetric fluidregion to the first volumetric fluid region being governed by a secondset of constraints that differs from the first set of constraints, thefirst set of constraints permitting a fraction of elements oriented tomove from the first volumetric fluid region to the second volumetricfluid region to move from the first volumetric fluid region 1 to thesecond volumetric fluid region, the fraction corresponding to a porosityof the at least one porous material being simulated.
 20. The method ofclaim 19, wherein the second set of constraints permits all elementsoriented to move from the second volumetric fluid region representingthe portion of physical or mechanical device to the first volumetricfluid region representing the at least one porous material to actuallymove from the second volumetric fluid region to the first volumetricfluid region.